A115052 Expansion of 1/(3*x^2 - 3*x + 1)^2.

1, 6, 21, 54, 108, 162, 135, -162, -1053, -2916, -5832, -8748, -8019, 4374, 41553, 118098, 236196, 354294, 334611, -118098, -1476225, -4251528, -8503056, -12754584, -12223143, 3188646, 49424013, 143489070, 286978140, 430467210, 416118303, -86093442, -1592728677

Offset: 0

Author

Roger L. Bagula, Feb 28 2006

Keywords

• sign
• easy
• changed

Comments

q=1 coefficient expansion of hierarchical lattice renormalization polynomial.

References

• Heinz-Otto Peitgen and Peter Richter (editors), The Beauty of Fractals, Springer-Verlag, New York, 1986, p. 146.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (6,-15,18,-9).

Crossrefs

Autoconvolution of A057083.

Programs

Magma

I:=[1,6,21,54]; [n le 4 select I[n] else 6*Self(n-1)-15*Self(n-2)+18*Self(n-3)-9*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 20 2011

Maple

A115052 := proc(n) 1/(3*x^2-3*x+1)^2 ; coeftayl(%,x=0,n) ; end proc: # R. J. Mathar, Sep 17 2011

Formula

From Stefano Spezia, Sep 01 2025: (Start)

a(n) = 3^(n/2)*((1 - n)*cos(n*Pi/6) + sqrt(3)*(3 + n)*sin(n*Pi/6)).

E.g.f.: exp(3*x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*(3 + 2*x)*sin(sqrt(3)*x/2)). (End)